From imaging to material identification: A generalized concept of topological sensitivity

doi:10.1016/j.jmps.2006.07.009

Journal of the Mechanics and Physics of Solids

Volume 55, Issue 2, February 2007, Pages 245-279

https://doi.org/10.1016/j.jmps.2006.07.009 Get rights and content

Abstract

To establish a compact analytical framework for the preliminary stress-wave identification of material defects, the focus of this study is an extension of the concept of topological derivative, rooted in elastostatics and the idea of cavity nucleation, to 3D elastodynamics involving germination of solid obstacles. The main result of the proposed generalization is an expression for topological sensitivity, explicit in terms of the elastodynamic Green's function, obtained by an asymptotic expansion of a misfit-type cost functional with respect to the nucleation of a dissimilar elastic inclusion in a defect-free “reference” solid. The featured formula, consisting of an inertial-contrast monopole term and an elasticity-contrast dipole term, is shown to be applicable to a variety of reference solids (semi-infinite and infinite domains with constant or functionally graded elastic properties) for which the Green's functions are available. To deal with situations when the latter is not the case (e.g. finite reference bodies or those with pre-existing defects), an adjoint field approach is employed to derive an alternative expression for topological sensitivity that involves the contraction of two (numerically computed) elastodynamic states. A set of numerical results is included to demonstrate the potential of generalized topological derivative as an efficient tool for exposing not only the geometry, but also material characteristics of subsurface material defects through a local, point-wise identification of “optimal” inclusion properties that minimize the topological sensitivity at sampling location. Beyond the realm of non-invasive characterization of engineered materials, the proposed developments may be relevant to medical diagnosis and in particular to breast cancer detection where focused ultrasound waves show a promise of superseding manual palpation.

Introduction

Elastic-wave identification of defects and heterogeneities embedded in semi-infinite or finite bodies is a problem of considerable interest in mechanics owing to its applications in material characterization, seismology, and medical diagnosis. The underlying inverse solutions can be derived from a variety of computational platforms that include e.g. (far-field) ray theory (Aki and Richards, 2002), finite-difference approximation of the wave equation (Sheriff and Geldart, 1995, Schroeder et al., 2002), and boundary integral formulations (Bonnet, 1995). In the context of 3D material characterization, these approaches carry a substantial computational cost associated with solving the forward problem. This precludes the use of global search techniques such as genetic algorithms which involve a large number of forward simulations. To mitigate the problem, gradient-based optimization algorithms have been proposed as a computationally tractable alternative to solving inverse scattering problems, especially when aided by the analytical gradient estimates (Plessix et al., 1998, Guzina and Bonnet, 2004). Unfortunately, the latter class of solutions necessitate a reliable preliminary information about the geometry and material characteristics of hidden defects for satisfactory performance.

Building on the results in shape optimization obtained for Laplace (Sokolowski and Zochowski, 1999, Garreau et al., 2001) and Helmholtz (Samet et al., 2004, Pommier and Samet, 2005) systems, Guzina and Bonnet (2004), Bonnet and Guzina (2004), and Gallego and Rus (2004) have recently established the method of topological sensitivity as a tool for preliminary, grid-based reconstruction of obstacles in the context of inverse elastic scattering that requires no prior information (or assumptions in the absence thereof) about the location and geometry of internal defects. In the approach the topological derivative, which quantifies the sensitivity of a given cost functional with respect to the nucleation of an infinitesimal obstacle in the reference (background) medium, is used as an effective obstacle indicator through an assembly of sampling points where it attains pronounced negative values. Typically, the formulas for topological sensitivity permit an explicit representation (e.g. in terms of the Green's function) that is responsible for the computational efficiency of this class of reconstruction techniques. Notwithstanding their usefulness, however, the foregoing analyses are limited in the sense that they are focused on the nucleation of impenetrable scatterers, and in particular cavities in 3D (Guzina and Bonnet, 2004, Bonnet and Guzina, 2004) and/or cracks in 2D (Gallego and Rus, 2004) elastodynamics.

In this study, the concept of topological sensitivity is generalized to permit the nucleation of dissimilar solid inclusions and thus allow for preliminary elastic-wave identification of subsurface defects of more general nature. On employing a boundary integral approach to derive the necessary asymptotics in terms of the vanishing defect size, it is shown that the proposed generalization (termed “material-topological” sensitivity) consists of a monopole term, related to the mass density contrast, and a dipole term involving the elasticity contrast between the defect and the matrix. To cater for engineering applications, explicit formulas are derived for canonical cases when the nucleating inclusion takes spherical or ellipsoidal shape. For generality, the proposed developments are recast within an alternative framework of the adjoint-field formulation that permits nucleation of inclusions in an arbitrary (infinite or finite, homogeneous or heterogeneous) reference solid. Through numerical examples it is shown that the material-topological sensitivity can be used, in the context of inverse scattering, as an effective defect indicator through an assembly of sampling points where it attains marked negative values. On varying the material characteristics of the nucleating obstacle, it is also shown that the featured sensitivity can be used as a preparatory tool for both geometric and material identification of internal defects.

Beyond their intrinsic potential for the study of localized damage evolution in natural and engineered materials (e.g. Mazars et al., 1991, Xu, 2004, Bonamy et al., 2005), the proposed developments be especially useful in breast cancer detection wherein the knowledge of the shear modulus of a lesion, (geometrically) identified via ultrasound or magnetic resonance imaging, may permit reliable differentiation between the malignant and benign growths (Sarvazyan et al., 1998, Fatemi and Greenleaf, 1998). For completeness, it is noted that the underlying idea of a nucleating inclusion, explored in this study, is similar in spirit to the recent work in Ammari and Kang (2004) dealing with Laplace, elastostatic and Helmholtz systems. Notwithstanding the apparent commonalities, however, the asymptotic expansion methodology, formulas for the generalized topological sensitivity, and the material-geometric identification approach proposed herein have not been established elsewhere.

Section snippets

Preliminaries

Consider the inverse scattering problem where the semi-infinite solid, probed by elastic waves, contains a bonded defect $Δ_{true}$ with smooth boundary $Γ_{true} = \partial Δ_{true}$ (see Fig. 1). With the Cartesian frame ${O; ξ_{1}, ξ_{2}, ξ_{3}}$ set at the top surface $Σ$ , the reference elastic half-space $Ω = {(ξ_{1}, ξ_{2}, ξ_{3}) | ξ_{3} > 0}$ with closure $\bar{Ω} = Ω \cup Σ$ is characterized by the shear modulus $μ$ , Poisson's ratio $ν$ , and mass density $ρ$ . Elastic parameters of the obstacle, $Δ_{true} \subset Ω$ , are $μ_{true}^{⋆}$ and $ν_{true}^{⋆}$ ; its mass density is denoted as $ρ_{true}^{⋆}$ .

Topological sensitivity

To aid the gradient-based minimization of (1) that is often used as a tool for identifying $Δ_{true}$ on the basis of motion measurements $u^{obs}$ , of interest in this study is the development of material-topological derivative for the class of cost functionals $J (Ω^{-}, m; f)$ given by (1) that would provide a reliable preliminary information about the location, geometry and material characteristics of the hidden defect. In situations when $J$ is a non-convex function in the material-geometric parametric space

Asymptotic for a nucleating inclusion

To arrive at a compact expression for (17) when $m$ is non-trivial, integral representation (12) when $Γ = S_{a}$ and $\tilde{u} = {\tilde{u}}^{a}$ can be conveniently rewritten by virtue of (10) as ${\tilde{u}}^{a} (x) = - e_{k} \int_{S_{a}} \hat{t}^{k} (ξ, x) \cdot u^{F} (ξ) d S_{ξ} - e_{k} \int_{S_{a}} \hat{t}^{k} (ξ, x) \cdot {\tilde{u}}^{a} (ξ) d S_{ξ} + e_{k} \int_{S_{a}} \hat{u}^{k} (ξ, x) \cdot t^{a} (ξ) d S_{ξ} \equiv I^{1} (x) + I^{2} (x) + I^{3} (x), x \in Ω^{-} = Ω ⧹ {\bar{B}}_{a},$ where $n$ denotes the unit normal on $S_{a} = \partial B_{a}$ oriented toward the interior of $B_{a}$ ; $t^{a} = n \cdot C : \nabla (u^{F} + {\tilde{u}}^{a})$ , and $I^{1}$ , $I^{2}$ , and $I^{3}$ denote, respectively, the three integrals on the right-hand side of (22). Note that the order of multiplicands

Direct formulation

By virtue of (10) and (49), one may expand (1) with respect to the scattered field caused by the vanishing obstacle, ${\tilde{u}}^{a} = u^{a} - u^{F}$ , as $J (Ω ⧹ {\bar{B}}_{a}, m; f) = J (Ω; f) + \int_{Π^{obs}} Re \{\frac{\partial ϕ}{\partial u} (u^{F}, u^{obs}) \cdot {\tilde{u}}^{a}\} d Π_{ξ} + o (∥ {\tilde{u}}^{a} ∥) as a \to 0,$ where $\frac{\partial ϕ}{\partial w} \equiv \frac{\partial ϕ}{\partial w_{R}} - i \frac{\partial ϕ}{\partial w_{I}}, w_{R} = Re (w), w_{I} = Im (w) .$ Since $\partial ϕ / \partial u (u^{F}, u^{obs})$ is by definition independent of a, one finds from (15), (49) and (50) that $h (a) \propto a^{3}$ for the 3D nucleating inclusion problem; a behavior that conforms with the hypothesis $\lim_{a \to 0} h (a) = 0$ made in (16). Although the choice of multiplicative constant in

Results and discussion

As shown in Guzina and Bonnet (2004) and Bonnet and Guzina (2004), the nucleating-cavity analogues of (54) and (74) can be used as a robust tool for the preliminary 3D reconstruction of impenetrable defects in the context of inverse scattering. Owing to their explicit dependence on the material properties of a nucleating inclusion, on the other hand, (54) and (74) carry an additional potential for material identification trough a parametric variation of $m$ geared toward minimizing the value of $T ($

Conclusions

In this study the concept of topological derivative, that has its origins in elastostatics and structural shape optimization, is extended to permit preliminary, yet robust 3D elastic-wave identification of material defects. In a departure from earlier studies that revolve around the idea of cavity nucleation, the proposed approach postulates the creation of an (infinitesimal) elastic inclusion. On taking the limit of a boundary integral representation of the scattered field caused by an elastic

Acknowledgments

The support provided by the National Science Foundation through Award No. CMS-0324348 to B. Guzina and the University of Minnesota Supercomputing Institute during the course of this investigation is kindly acknowledged.

References (39)

M. Bonnet
BIE and material differentiation applied to the formulation of obstacle inverse problems
Eng. Anal. with Bound. Elem.
(1995)
M. Bonnet
Regularized BIE formulations for first- and second-order shape sensitivity of elastic fields
Comput. Struct.
(1995)
B.B. Guzina et al.
Elastodynamic Green's functions for a smoothly heterogeneous half-space
Int. J. Solids Struct.
(1996)
B.B. Guzina et al.
On the stress-wave imaging of cavities in a semi-infinite solid
Int. J. Solids Struct.
(2003)
R.Y.S. Pak et al.
Three-dimensional wave propagation analysis of a smoothly heterogeneous solid
J. Mech. Phys. Solids
(1995)
R.Y.S. Pak et al.
Seismic soil-structure interaction analysis by direct boundary element methods
Int. J. Solids Struct.
(1999)
A.P. Sarvazyan et al.
Shear wave elasticity imaging: a new ultrasonic technology of medical diagnostics
Phys. Med. Biol.
(1998)
X.H. Xu
Damage evaluation and damage localization of rock
Theor. Appl. Fract. Mech.
(2004)
J.D. Achenbach
Reciprocity in Elastodynamics
(2003)
K. Aki et al.
Quantitative Seismology
(2002)

H. Ammari et al.

Reconstruction of the Small Inhomogeneities from Boundary Measurements

(2004)

D. Bonamy et al.

Nano-ductile crack propagation in glasses under stress corrosion: spatiotemporal evolution of damage in the vicinity of the crack tip

Int. J. Solids Struct.

(2005)

M. Bonnet

Boundary Integral Equation Methods for Solids and Fluids

(1999)

M. Bonnet et al.

Sounding of finite solid bodies by way of topological derivative

Int. J. Num. Meth. Eng.

(2004)

D. Colton et al.

Inverse Acoustic and Electromagnetic Scattering Theory

(1992)

G. Dahlquist et al.

Numerical Methods

(1974)

H.A. Eschenauer et al.

Bubble method for topology and shape optimization of structures

Struct. Optim.

(1994)

J.D. Eshelby

The determination of the elastic field of an ellipsoidal inclusion and related problems

Proc. Roy. Soc. London A

(1957)

M. Fatemi et al.

Probing the dynamics of tissue at low frequencies with the radiation force of ultrasound

Phys. Med. Biol.

(1998)

Cited by (85)

A fast directional boundary element method for wideband multi-domain elastodynamic analysis
2019, Engineering Analysis with Boundary Elements
A fast directional boundary element method for wideband multi-domain elastodynamic analysis is developed. The directional low rank property of the elastodynamic kernel is proved, which serves as the theoretical basis of the fast directional algorithm. The boundary integral evaluation is accelerated domain by domain. For each domain, the octree is divided into low frequency and high frequency regimes according to its S-wave number. The computation in the low frequency regime is the same with the kernel independent fast multipole method, while the high frequency regime is computed efficiently by applying the parabolic low rank property of the kernels on directional wedges. The matrices on only one directional wedge for each level have to be calculated, since that for other directions can be obtained by coordinate frame rotations. Thus the harmonic responses for any frequencies can be computed efficiently. Numerical results show that the computational complexity for wideband multi-domain elastodynamic problems is successfully brought down to O(Nlog^αN). Its applications to viscoelastic materials and inverse problems are also validated.
Solving inverse geometry heat conduction problems by postprocessing steady thermograms
2019, International Journal of Heat and Mass Transfer
This paper deals with the use of steady active thermographic inspection as a structural health monitoring method to detect defects and inclusions in a heat conductive medium. The proposed postprocessing method has the advantage of taking into account the whole physics of the problem instead of a reduced set of physical information, as most of the traditional methods do. In comparison with other non-destructive testing techniques like ultrasonics, thermographic inspection is non-contact, more non-intrusive, and safer. However, processing thermographic data is a fairly demanding problem because heat conduction is short range and exhibits a poor signal-to-noise ratio. Numerical experiments illustrate that the present postprocessing technique detects damage fairly well even in very challenging scenarios where the size of the defects is rather small and/or thermograms are highly noisy.
Detection of multiple impedance obstacles by non-iterative topological gradient based methods
2019, Journal of Computational Physics
We investigate a fast, one-step imaging method of multiple 2D and 3D acoustic obstacles fully-coated by a complex surface impedance with either monochromatic or multi-frequency noisy data. Introducing the topological gradient of the misfit functional as a limit of shape derivatives, closed-form expressions of the obstacle indicator are derived using Fourier and Mie series expansions of the radiating solution. We provide a wide variety of numerical experiments that assesses the performance and limitations of the one step single and multi-frequency imaging strategies when dealing both with full and limited aperture measurements.
Topological sensitivity based far-field detection of elastic inclusions
2018, Results in Physics
The aim of this article is to present and rigorously analyze topological sensitivity based algorithms for detection of diametrically small inclusions in an isotropic homogeneous elastic formation using single and multiple measurements of the far-field scattering amplitudes. A $L^{2}$ -cost functional is considered and a location indicator is constructed from its topological derivative. The performance of the indicator is analyzed in terms of the topological sensitivity for location detection and stability with respect to measurement and medium noises. It is established that the location indicator does not guarantee inclusion detection and achieves only a low resolution when there is mode-conversion in an elastic formation. Accordingly, a weighted location indicator is designed to tackle the mode-conversion phenomenon. It is substantiated that the weighted function renders the location of an inclusion stably with resolution as per Rayleigh criterion.
A level-set-based topology optimisation for acoustic–elastic coupled problems with a fast BEM–FEM solver
2017, Computer Methods in Applied Mechanics and Engineering
This paper presents a structural optimisation method in three-dimensional acoustic–elastic coupled problems. The proposed optimisation method finds an optimal allocation of elastic materials which reduces the sound level on some fixed observation points. In the process of the optimisation, configuration of the elastic materials is expressed with a level set function, and the distribution of the level set function is iteratively updated with the help of the topological derivative. The topological derivative is associated with state and adjoint variables which are the solutions of the acoustic–elastic coupled problems. In this paper, the acoustic–elastic coupled problems are solved by a BEM–FEM coupled solver, in which the fast multipole method (FMM) and a multi-frontal solver for sparse matrices are efficiently combined. Along with the detailed formulations for the topological derivative and the BEM–FEM coupled solver, we present some numerical examples of optimal designs of elastic sound scatterer to manipulate sound waves, from which we confirm the effectiveness of the present method.
Cavity identification in elastic structures by explicit domain mapping and boundary mode sensitivity analysis
2019, European Journal of Mechanics, A/Solids
Citation Excerpt :
Over the years, several solution strategies have been developed to overcome the difficulty and they are presented as follows, The MIP(material identification problem) strategy where the GIPs are equivalently treated as the MIPs (Guzina and Chikichev, 2007; Bendsoe, 1989; Cea et al., 2000). For a domain with interior cavities or inclusions, one can always replace the cavities or the inclusions of the original domain by distributive material coefficients which tend to zero for a cavity and to infinity for a rigid inclusion.
This paper presents a novel cavity identification approach for linear elastic structures with the boundary mode data. The proposed approach consists of two major aspects. The first is the construction of an explicit domain mapping that maps the reference domain onto the physical domain by the level set functions. In proceeding so, the R-functions theory not only provide an analytical methodology to get the level set functions for the external and interior cavity boundaries, but also construct the desired domain mapping directly by level-set-functions weighted interpolation of the parameterized boundary mappings. The physical domain with the cavities is explicitly parameterized in the domain mapping and the desired mesh keeps fixed over the reference domain. The second is the enhanced sensitivity-based approach for minimum solution to the nonlinear least-squares goal function of the cavity identification problem. The sensitivity of the boundary modes to the geometric parameters is analyzed with the help of the acquired domain mapping and the trust-region constraint is introduced in order to reach a weak convergence. Numerical examples are conducted to show the feasibility and efficiency of the present cavity identification approach.

View all citing articles on Scopus

View full text

From imaging to material identification: A generalized concept of topological sensitivity

Abstract

Introduction

Section snippets

Preliminaries

Topological sensitivity

Asymptotic for a nucleating inclusion

Direct formulation

Results and discussion

Conclusions

Acknowledgments

Eng. Anal. with Bound. Elem.

Comput. Struct.

Int. J. Solids Struct.

Int. J. Solids Struct.

J. Mech. Phys. Solids

Int. J. Solids Struct.

Phys. Med. Biol.

Theor. Appl. Fract. Mech.

Reciprocity in Elastodynamics

Quantitative Seismology

Reconstruction of the Small Inhomogeneities from Boundary Measurements

Nano-ductile crack propagation in glasses under stress corrosion: spatiotemporal evolution of damage in the vicinity of the crack tip

Int. J. Solids Struct.

Boundary Integral Equation Methods for Solids and Fluids

Sounding of finite solid bodies by way of topological derivative

Int. J. Num. Meth. Eng.

Inverse Acoustic and Electromagnetic Scattering Theory

Numerical Methods

Bubble method for topology and shape optimization of structures

Struct. Optim.

The determination of the elastic field of an ellipsoidal inclusion and related problems

Proc. Roy. Soc. London A

Probing the dynamics of tissue at low frequencies with the radiation force of ultrasound

Phys. Med. Biol.